# Hildreth-Lu Technique for Autocorrelation Correction¶

hl <rho1> <rho2> <incr> <y> = <x1>, [<x2> [, <x3> [, ... [, <xn>]]]]

In the Hildreth-Lu procedure, a guess of the autocorrelation coefficient of the errors, rho, is chosen, multiplied by the equation lagged once, and subtracted from the unlagged equation. The resulting equation then is estimated by ordinary regression. Another value of the guess of rho then is chosen and the process repeated. In this command, <rho1> is the starting guess of rho, <incr> is the amount by which it is incremented on each iteration, and <rho2> is an upper limit on the guess. The <y> and <x1>, ..., <xn> values are as in the r command.

At the end of the process, you will get a table with this heading:

```RHO-HL    SEE 1 AHEAD  RHO-EST    SEE LONG-RUN
```

RHO-HL shows the assumed rho, the SEE 1 AHEAD shows the standard error of estimate (SEE) of the estimated equation, RHO-EST shows the rho of the estimated equation, and SEE LONG-RUN shows the standard error using the fitted equation on the original, undifferenced data, without a knowledge of the true lagged value of the dependent variable, as must be done in forecasts of more than a few periods ahead. If the save command is on, all of the estimated equations will placed in the .SAV file as undifferenced equations suitable for going into a model. You must choose which one you want.

For graphing the HL output, one should run the equation one last time with the chosen value of rho as both the starting and ending rho. Then also run the equation with just the r command (no autocorrelation). Finally, construct two graphs:

```gr depvar predp1 hlshort
gr depvar predic hllong
```

The first will compare the actual with two one-period ahead forecasts, the one from ordinary least squares (+’s) and one from Hildreth-Lu (x’s). The second compares the actual with two forecasts not relying on the lagged value of the dependent variable. Again, the uncorrected least squares fit is shown by +’s and the fit using the H-L correction is shown by x’s.

The pauses at the end of each fit in the HL procedure can be eliminated by the zip command. If you use zip, however, you must remember to do “zip off” before using the (gr)aph command.

G7 does not provide the Corchrane-Orcutt correction, since Hildreth-Lu is better.

Regression Tests

ARIMA Techniques